For every Lagrangian fibration of a projective hyper-Kähler manifold, the perverse filtration equals the monodromy weight filtration of an associated type III degeneration.
Torus fibers and the weight filtration
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abstract
We show that if $(X,Y)$ is a simple normal crossings log Calabi--Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of $Y$ which can be used to construct $W_{2k-1}H^k(X \setminus Y;\mathbb{Q})$ for all $k$. We show that an analogous result holds for degenerations of Calabi--Yau varieties. We use this to show that P=W type results hold for pairs $(X,Y)$ consisting of a rational surface $X$ and a nodal anticanonical divisor $Y$, and for K3 surfaces.
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P=W for Lagrangian fibrations and degenerations of hyper-K\"ahler manifolds
For every Lagrangian fibration of a projective hyper-Kähler manifold, the perverse filtration equals the monodromy weight filtration of an associated type III degeneration.